Let O be the vertex and Q be any point on the parabola, x ^ 2 = 8y...

Let $P\left(h, k\right)$ divide $OQ$ in ratio $1: 3$

\left(h, k\right) = \left(\dfrac{1\left(4t\right) + 3\left(0\right)}{1 + 3}, \dfrac{1\left(2{t}^{2}\right) + 3\left(0\right)}{1 + 3}\right)$</p><p>\left(h , k\right) = \left(t,\dfrac{2{t}^{2}}{4}\right)$

$h = t \text{ and } k = \dfrac{{t}^{2}}{2}$

$k = \dfrac{{h}^{2}}{2}$

$⇒ {x}^{2} = 2y$ is the required locus.